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We initiate the study of a class of noncommutative domains of n-tuples ofbounded linear operators on a Hilbert space, which is generated by certainpositivity conditions on polynomials in n noncommutative indeterminates. Weobtain Fatou type results, functional calculi, and a model theory for n-tuplesof operators in these domains. The main role in this study is played by a classof noncommutative Berezin transforms which is introduced in this paper. Ourresults extend to noncommutative varieties.

Noncommutative Berezin transforms and model theory